Oil wells typically have multiple concentric pipes called casing strings. In FIG. 1, the configuration 100 of two concentric pipes is illustrated. The internal pipe 102 is designated “tubing” and the external pipe 104 is designated “casing.” There is a wellbore 106 that is considered rigid in this analysis.
For a set of two concentric strings, if the internal pipe has a compressive axial force, it will typically deform into a helically shaped configuration within the other string, as shown in FIG. 1. The cross-sectional areas of the various pipes are described by:Ati=πrti2 Ate=πrte2 Aci=πrci2 Ace=πrce2  (1)where rti is the inside radius of the tubing, rte is the outside radius of the tubing, rci is the inside radius of the casing, and rce is the outside radius of the casing. Clearances between the various pipes and the wellbore are given as:rc=rci−rte roc=rw−rce  (2)
Where rc is the radial clearance between the tubing and casing, and roc is the radial clearance between the casing and the wellbore and rw is the wellbore radius. Most analyses of this problem assume that the outer casing is rigid. In reality, this external casing is also elastic and would displace due to the loads generated by contact with the internal pipe. Further, if both strings have compressive axial forces, both strings will buckle, and the resulting buckled configuration must fit together so that contact forces between the two strings are positive and the pipes do not each occupy the same space. If the two strings have an external, cylindrical rigid wellbore, then any contact forces with this wellbore must also be positive and the buckled pipe system must lie within this wellbore. This configuration is illustrated as a cross-section in FIG. 1 before buckling takes place. The post-buckling configuration 200 is illustrated in FIG. 2.
There is only one known solution to the problem presented by multiple concentric buckling pipes, which is described in SPE 6059 by Stan A. Christman entitled “Casing Stresses Caused by Buckling of Concentric Pipes.” In this article, a composite pipe based on the summed properties of the individual pipes is proposed. Further, the pipes do not touch each other, but are assumed to remain concentric. The deficiency in this analysis is that it does not conform to the requirements that i) the contact forces between the two strings are positive and the pipes do not each occupy the same space; and ii) the contact forces with the wellbore are positive and the buckled pipe system lies within the wellbore. As a result the assumption that the pipes do not touch each other but remain concentric renders an inaccurate displacement solution.